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**nqramjets** Okay, this may be a slight difference of definition, but here is the issue:

A function is $\displaystyle f:[0,1]\rightarrow \mathbb{R}$ is Riemann Integrable on $\displaystyle [0,1]$ if there is an $\displaystyle L\in \mathbb{R}$ such that for all $\displaystyle \varepsilon >0$ there exists $\displaystyle \delta >0$ such that for every tagged partition $\displaystyle \dot{P}$ of $\displaystyle [0,1]$ with $\displaystyle \|\dot{P}\|<\delta$, then

$\displaystyle |S(f;\dot{P})-L|<\varepsilon$

Now, if you look at the Riemann sum part you have

$\displaystyle S(f;\dot{P})=\sum_{i=1}^nf(t_i)(x_i-x_{i-1})$ but if I choose my tag of the first sub-interval to be 0, ie. $\displaystyle t_1=0\in [x_0,x_1]$ then $\displaystyle f(t_i)=\frac{1}{\sqrt{t_i}}=\frac{1}{0}$ which is undefined. So, $\displaystyle f\not\in R[0,1]$ and therefore we cannot apply the FTC.

$\displaystyle \frac{1}{\sqrt{x}}$ simply *is not* a function from $\displaystyle [0,1]$ into $\displaystyle \mathbb{R}$ because 0 is not in the domain.

What is your definition?